# How to use the distributive law correctly in propositional logic?

Can someone explain how in propositional logic these are equivalent :

A ∧ B ∧ (¬B ∨ ¬C) ≡ A ∧ B ∧ ¬C


Because using the distributive law I would get:

 ≡ (A ∧ B ∧ ¬B) ∨ (A ∧ B ∧ ¬C)
≡  A ∨ (A ∧ B ∧ ¬C)
≡ (A ∨ A) ∧ (A ∨ B) ∧ (A ∨ ¬C)
≡  A ∧ (A ∨ B) ∧ (A ∨ ¬C)


## 2 Answers

Hint. B ∧ ¬B is false, so what is A ∧ B ∧ ¬B ?

• Oh I see, so A ∧ B ∧ ¬B must be false , leaving just A ∧ B ∧ ¬C – xava May 23 at 9:16
• Well done, you got it. – J.-E. Pin May 23 at 9:24

Your first use of the distributive law was correct. Now note that the statement $$A\wedge B\wedge \neg B$$ is a contradiction, and therefore we have $$\begin{array}{rcl}A\wedge B \wedge (\neg B \vee \neg C) &\equiv& (A\wedge B \wedge \neg B) \vee (A \wedge B \wedge \neg C)\\&\equiv &(A\wedge (B \wedge \neg B)) \vee (A \wedge B \wedge \neg C)\\ &\equiv& (A\wedge F)\vee (A \wedge B \wedge \neg C)\\& \equiv&F \vee (A \wedge B \wedge \neg C)\\&\equiv& (A \wedge B \wedge \neg C) \end{array}$$ Where $$F$$ denotes falsem.